Research Problems in Harmonic Analysis and Partial Differential Equations

调和分析与偏微分方程的研究问题

• 批准号: 1160981
• 负责人: Shuanglin Shao
• 金额: $ 9.98万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1160981&HistoricalAwards=false
• 关键词: Research; Problems; Harmonic; Analysis; Partial

The investigator plans to investigate the extremisers problem for the Tomas-Stein inequality for the sphere, to establish a Strichartz estimate for an oscillatory integral with a non-elliptic phase, and to investigate the Cauchy problem for the two dimensional water wave equations with surface tension. The first two projects aim to understand some aspects of oscillatory integrals, which are important in the restriction/extension theory of Fourier transforms to hyper-spaces with curvature in the Euclidean spaces. The extremisers problem for the Tomas-Stein inequality asks whether there exists a function which optimizes the inequality so that it becomes an equality. It also includes questions of characterizing extremisers such as establishing the smoothness property. The first project concerns the extremisers problem for the Tomas-Stein inequality for the one and higher dimensional spheres. The second project focuses on studying an oscillatory integral with a non-elliptic phase, which is in form of Strichartz estimates. It is motivated by the applications of these estimates in the Schrodinger equations. The third project is to investigate the Cauchy problem for the two dimensional water wave equations with surface tension. It is well known that an important step in approaching this problem in previous works is a reduction of the original system of equations to a suitable and equivalent quasilinear system. So it remains an interesting question how such a reduction will play a role in developing its wellposedness theory.The proposed research will result in broader impact from several points of view. First, these problems under investigation lie at the interface of several branches of mathematics, e.g. analysis and partial differential equations. Thus their solutions will facilitate interactions among these fields. Moreover, these problems concern both the development of mathematical techniques and the human understanding of the fundamental concepts in mathematics. For instance, the Tomas-Stein inequality is an important measure of properties of a basic operation in mathematics, the Fourier transform, and goes back to a fundamental question: when is an infinite series (in this case the Fourier series of a given function) summable? Examples of ways in which these problems impact applied science and engineering are many and varied. For instance, the water wave equation is used as a model to describe the propagation of surface waves on a river or the ocean. A rigorous study of this model will provide the theoretical ground for modeling and computational simulations, which in turn allow researchers to gain some insight into some destructive physical phenomena such as the rogue waves and tsunamis. Finally, sharing the research experience through the investigator's teaching activities will increase the awareness and appreciation of mathematics research, and improve diversity of the scientific community and the society.

研究者计划研究球的thomas - stein不等式的极值问题，建立非椭圆相位振荡积分的Strichartz估计，并研究具有表面张力的二维水波方程的Cauchy问题。前两个项目旨在理解振荡积分的某些方面，这在欧几里德空间中具有曲率的超空间的傅里叶变换的限制/扩展理论中是重要的。thomas - stein不等式的极值问题问的是是否存在一个函数使不等式最优化从而使它成为一个不等式。它还包括表征极值的问题，如建立平滑性。第一个项目涉及一维和高维球体的托马斯-斯坦不等式的极值问题。第二个项目的重点是研究非椭圆相位的振荡积分，其形式为Strichartz估计。它的动机是这些估计在薛定谔方程中的应用。第三个课题是研究具有表面张力的二维水波方程的柯西问题。众所周知，在以前的工作中，解决这个问题的一个重要步骤是将原来的方程组简化为一个合适的、等价的拟线性系统。因此，这种减少将如何在发展其适位性理论中发挥作用仍然是一个有趣的问题。从几个角度来看，拟议的研究将产生更广泛的影响。首先，这些正在研究的问题处于几个数学分支的交叉点，例如分析和偏微分方程。因此，他们的解决方案将促进这些领域之间的相互作用。此外，这些问题既关系到数学技术的发展，也关系到人类对数学基本概念的理解。例如，托马斯-斯坦不等式是数学中基本运算傅里叶变换性质的重要度量，它回到了一个基本问题：无穷级数（在这种情况下是给定函数的傅里叶级数）何时是可和的？这些问题影响应用科学和工程的方式有很多，也有很多不同的例子。例如，水波方程被用作描述河流或海洋表面波传播的模型。对该模型的严格研究将为建模和计算模拟提供理论基础，从而使研究人员能够深入了解一些破坏性的物理现象，如巨浪和海啸。最后，通过研究者的教学活动分享研究经验，将增加对数学研究的认识和欣赏，提高科学界和社会的多样性。